Mathematics • Year 8 • Unit 4 • Lesson 6
Line Graphs
Build fluency reading and constructing line graphs, describing trends, and distinguishing interpolation from extrapolation. One worked example, one guided example with blanks, then eight independent problems.
1. I do — fully worked example
Read every line. Each step has a short reason so you can see why we do it, not just what we do.
Problem. Monthly rainfall (mm): Jan 80, Feb 65, Mar 55, Apr 40, May 25, Jun 15. Describe the trend, then estimate the rainfall in mid-April using interpolation. State the reliability.
Step 1 — Plot the points and describe the trend.
Plot (Jan, 80), (Feb, 65), (Mar, 55), (Apr, 40), (May, 25), (Jun, 15) on axes time vs rainfall.
Trend: values fall every month → DECREASING trend.
Reason: connect consecutive points; the overall direction is the trend.
Step 2 — Estimate mid-April rainfall.
Mid-April sits halfway between Apr (40 mm) and May (25 mm). Average ≈ (40 + 25) ÷ 2 = 32.5 mm.
Reason: mid-April lies WITHIN the known data range — this is interpolation.
Step 3 — Classify and judge reliability.
This is INTERPOLATION (within the data range) → generally RELIABLE.
Answer: Trend = decreasing. Mid-April ≈ 32 mm. Interpolation — reliable.
2. We do — fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. A line graph shows the temperature (°C) at 6am, 8am, 10am, 12pm: 12, 15, 20, 27. Describe the trend, estimate the temperature at 9am, and state whether your estimate is interpolation or extrapolation.
Step 1 — Trend:
Values go from 12 → 15 → 20 → 27, so the trend is ______ (increasing / decreasing / constant).
Step 2 — Estimate at 9am (halfway between 8am = 15°C and 10am = 20°C):
Average ≈ (15 + 20) ÷ 2 = ______°C
Step 3 — Classify:
9am lies ______ the data range (within / outside), so this is ______ (interpolation / extrapolation).
Step 4 — Reliability:
The estimate is ______ (reliable / less reliable).
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation. The middle two are standard. The last two are extension.
Foundation — quick recall
3.1 A line graph shows annual sales: $120k, $135k, $148k, $162k, $178k over 5 years. Describe the trend in one word. 1 mark
3.2 Define interpolation and extrapolation in one sentence each. 1 mark
3.3 Which of these data sets is most suitable for a line graph?
(a) Favourite sports of 30 students.
(b) Eye colours of a classroom.
(c) Daily maximum temperature over two weeks. 1 mark
3.4 List THREE required features of a correctly drawn line graph. 1 mark
Standard — two-step problems
3.5 A line graph shows temperature at 1pm = 28°C and at 5pm = 32°C, joined by a straight line. (a) Estimate the temperature at 3pm. (b) Is this interpolation or extrapolation? 2 marks
3.6 A graph shows monthly sales: Jan 120, Feb 135, Mar 150, Apr 165, May 180, Jun 195 (units). (a) State the constant monthly increase. (b) Estimate July sales by extrapolation. (c) Comment on reliability. 2 marks
Extension — analyse the graph
3.7 A line graph shows a town's population at 10-year intervals: 1990: 8 000, 2000: 11 000, 2010: 15 000, 2020: 20 000. (a) Estimate the population in 2005 (interpolation). (b) Estimate the population in 2030 (extrapolation). (c) Which estimate is more reliable, and why? 2 marks
3.8 A student draws a line graph of favourite colours (Red, Blue, Green) connecting the bars with straight lines. Explain in one or two sentences why this is mathematically meaningless, and state the correct chart type. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (morning temperatures)
Step 1: Trend = increasing.
Step 2: (15 + 20) ÷ 2 = 17.5°C.
Step 3: 9am is within the data range → interpolation.
Step 4: Estimate is reliable.
3.1 — Trend description
Increasing.
3.2 — Definitions
Interpolation = estimating a value within the known data range (generally reliable). Extrapolation = predicting a value beyond the known data range (less reliable).
3.3 — Most suitable for a line graph
(c) Daily maximum temperature over two weeks — continuous numeric data over time. (a) and (b) are categorical → bar chart.
3.4 — Required features
Any three: descriptive title; both axes labelled with variable AND units; even scale on each axis; dot at each data point; lines connecting consecutive points.
3.5 — Temperature at 3pm
(a) 3pm is halfway between 1pm (28°C) and 5pm (32°C). Estimate ≈ (28 + 32) ÷ 2 = 30°C.
(b) Interpolation (within the data range).
3.6 — Monthly sales
(a) Monthly increase = 15 units (135−120, 150−135, etc., all equal to 15).
(b) July ≈ 195 + 15 = 210 units.
(c) This is extrapolation — less reliable than interpolation. The trend may not continue (e.g. summer slowdown could reduce sales).
3.7 — Town population
(a) 2005 sits halfway between 2000 (11 000) and 2010 (15 000) → estimate ≈ 13 000. Interpolation.
(b) 2030 is beyond the data range. If the trend continues, estimate ≈ 26 000. Extrapolation.
(c) The 2005 estimate (interpolation) is more reliable — extrapolation to 2030 assumes the growth pattern continues unchanged for another 10 years, which may not happen (could plateau, accelerate, or reverse).
3.8 — Favourite colours
Connecting points with a line implies that intermediate values exist between Red and Blue, which is nonsense for categorical data — there is no value "halfway between" Red and Blue. The correct chart is a bar chart, which keeps each category as a separate bar with gaps between them.